If a resident of Beverly Hills commits a serious offence, such as murder, in Los Angeles, where LAPD has jurisdiction, and then returns to his or her Beverly Hills' home, can LAPD officers legally arrest the perpetrator or must the matter be handled by Beverly Hills' police officers?
Yes, the Los Angeles PD officers can arrest you at your home.
An officer in California may arrest a person outside the geographical boundaries of their primary jurisdiction if the crime was committed in their jurisdiction, exigent circumstances exist, or they have consent from the agency having jurisdiction where the arrest occurred. (Hamilton (1986) 191 Cal.App.3d Supp. 13.)
In addition, the a law enforcement officer in CA can make an arrest for any public offense committed in their presence even if outside their primary jurisdiction.
However, Los Angeles and most every other agency in the state grants written permission to all agencies within the state to enforce state laws within their jurisdictional boundaries. It is common practice for a new agency head to send a notice to all agencies in the state granting that agency permission to act within its jurisdictional boundaries. In addition, many counties - Los Angeles included (at least as of about 7 years ago and for many years before that) - have an interagency MOU in place to further cement this process and allow for such an arrest.
In practice, the LAPD would likely notify BHPD that they were enroute to the residence to make the arrest and would advise if they need any assistance. It is likely that the BHPD would send someone to accompany them, but it is not a requirement.